Factual error: An An-12, the aircraft the main character uses to fly all over the world, has a maximum range of about 3,500 miles. Hardly enough to fly from the US to South Korea or from South Korea to Israel. The An-12 also miraculously transforms into a C-130 in a couple of filler scenes. And why is this ex-Soviet aircraft marked in USAF markings, assigned to McGuire AFB?
Other mistake: As Gerry enters the aircraft carrier command centre, we see screens with "world projections" of remaining uninfected populations by continent. They are counting down (illustrating the quick spread of the disease). However, the countdown speed of all of them, is approximately 3% each second - which means humankind would have no more than 20 seconds at this rate. The countdown speed should realistically be about 100,000 times slower than that (about 3% each day), which would still be the fastest pandemic spread the world has ever seen. (00:30:15)
Suggested correction: The countdowns are only seen for a couple of seconds, so there's no evidence that the countdowns are constant at the same rate. They may be updated in bursts as data is received. They may just show a rapid initial reduction that slows then flattens out. They may be wrong (because how could they be correct). Lots of explanations why this might not be what it seems.
Factual error: When flying from Korea to Israel, a nuclear explosion happens in the distance, the plane is hit by a shockwave, and then the phones go dead. The EMP from the nuke would have taken out all electronics first, not last. And they certainly wouldn't work again, especially a sat phone designed to receive radio waves. It would have probably also taken down the plane.
Suggested correction: There's no evidence this was a nuclear explosion. The mushroom cloud effect is most commonly associated with a nuclear explosion, but any sufficiently energetic detonation or deflagration will produce the same effect.
Except in this case, the explosion shown is a copy of a real nuclear detonation (Operation Crossroads, underwater), so it's obviously meant to be a nuclear explosion.
Plot hole: The same zombie who can smell through a vault that Gerry is not worth a bite, just minutes before couldn't smell the presence of three alive humans two meters apart.
Suggested correction: It's not until the zombie is in the vault with Gerry, after Gerry is infected, that it can smell that he "isn't worth a bite".
Other mistake: Despite the number of staff and zombies (former staff) at the W.H.O Research Centre, the car park leading up to the main entrance is empty but for one car.
Suggested correction: Maybe the main staff car park was round the back.
Factual error: The first time Brad Pitt leaves the aircraft carrier he is on an Antonov An-12. I haven't been able to find any evidence of an Antonov An-12 being able to take off or land on an aircraft carrier. The minimum runway distance for takeoff is 1500 m, and A Nimitz class carrier is only 325 m. (01:22:00)
Suggested correction: Most planes, if not all, that don't have vertical takeoff capabilities can't land or take off on aircraft carriers. That's why systems are in place to launch planes up to speed for take off and to catch or stop planes upon landing. Even jets that are 5 times faster than a An-12 need help with take off and landing.
A C-130 did multiple take-offs and landings aboard a carrier WITHOUT any assistance, so yes, an AN-12 might have been able to do the same thing: https://theaviationgeekclub.com/the-story-of-how-the-c-130-hercules-became-the-biggest-aircraft-to-land-on-an-aircraft-carrier/.
Other mistake: When approaching Israeli airspace the ATC refuses to land Gary's aircraft, but in real life an UN investigator approaching a country to meet its intelligence officer would have pre planned. They don't have to beg to land the plane.
Suggested correction: Standard "real life" procedures may not always apply in a zombie apocalypse.
Factual error: When we are at the Welsh WHO facility we get an exterior shot during the night where we hear crickets chirping. This is a classic US film-making error. There are no crickets in Wales or any part of the UK. What you would hear at night is silence. Maybe owls.
Suggested correction: What on earth makes you think there are no crickets in the UK? The most common type found in the UK is the field cricket (Gryllus campestris). In total there are over 25 types of crickets in the UK. Ture that some species are mute, but not the field cricket.https://en.wikipedia.org/wiki/Gryllus_campestris.
Crickets in Wales don't make a noise as they do in warmer climates.
The North Wales Wildlife Trust seems to disagree with you: "The Dark bush-cricket can be found in South Wales. Its irregular chirpings are a familiar sound of summer." https://www.northwaleswildlifetrust.org.uk/wildlife-explorer/invertebrates/grasshoppers-and-crickets/dark-bush-cricket.
Suggested correction: Can't speak to the second half of your paragraph (should really post as 3 separate mistakes) but as for the first, a range of 3500 miles, aircraft such as the kc-135 exist and aerial refueling is fairly common place. Considering it's a mission supported by the acting UN Secretary General to stop a world crisis, resources could have been diverted for refueling.
The initial launch from the carrier is a C-130 which can do this (if empty, minimal fuel, has the full length of the flight deck and the carrier is steaming full ahead into the wind). It then morphs into an AN-12 and back to a Hercules. They make the point that this small fleet is what is known to remain of allied forces so not sure where any tanker support will come from. Many movies have ridiculous range issues with aircraft anyway.